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Describing Groups [article]

Meng-Che Ho
<span title="2016-04-15">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study two complexity notions of groups - a computable Scott sentence and the index set of a group.  ...  In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups, generalizing methods that was previously used by J. Knight et al.  ...  Special thanks to Uri Andrews and to Tullia Dymarz for being such supportive and fantastic advisors.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.03729v3">arXiv:1512.03729v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qmyfooov3nblbpohst56vxlbqi">fatcat:qmyfooov3nblbpohst56vxlbqi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200903140223/https://arxiv.org/pdf/1512.03729v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9f/1f/9f1ffdb115efbe313686961b1f7e58b5cb8db8b3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.03729v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Index Sets of Computable Structures [article]

Wesley Calvert, Valentina S. Harizanov, Julia F. Knight, Sara Miller
<span title="2008-03-22">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The form of the sentence (computable Π_n, d-Σ_n, or Σ_n) yields a bound on the complexity of the index set. When we show m the index set, we know that the sentence is optimal.  ...  The index set of a computable structure A is the set of indices for computable copies of A.  ...  There is a computable Σ 3 Scott sentence, and the index set is mcomplete Σ 0 3 . Often, the first Scott sentence that comes to mind is not optimal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0803.3294v1">arXiv:0803.3294v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/aoinimw3yjdknbvaodjidhchfa">fatcat:aoinimw3yjdknbvaodjidhchfa</a> </span>
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Index sets of computable structures

W. Calvert, V. S. Harizanov, J. F. Knight, S. Miller
<span title="">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nu3fafenofagfe6orfxnkvy6oi" style="color: black;">Algebra and Logic</a> </i> &nbsp;
The index set of a computable structure A is the set of all indices for computable isomorphic copies of A.  ...  We determine, using the arithmetical hierarchy and the difference hierarchy, the exact complexity of the index sets of structures within the following classes of structures: finite structures, vector spaces  ...  There is a computable Σ 3 Scott sentence, and the index set is m-complete Σ 0 3 . Often the first Scott sentence that comes to mind is not optimal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10469-006-0029-0">doi:10.1007/s10469-006-0029-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dfsp5pwpcvdcrcjbtqtpqcrhm4">fatcat:dfsp5pwpcvdcrcjbtqtpqcrhm4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20060903201806/http://home.gwu.edu/~harizanv/SubIndexSets.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e4/d2/e4d2c6b04f6d5d99bffef08b73f0dd51c0012bb7.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10469-006-0029-0"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Complexity of Scott Sentences [article]

Rachael Alvir, Charles McCoy, Julia Knight
<span title="2018-07-07">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that if a countable structure A has a computable Σ_α Scott sentence and one that is computable Π_α, then it has one that is computable d-Σ_β for some β < α.  ...  We give effective versions of some results on Scott sentences.  ...  set whose index e i (according to some canonical indexing of the c.e. sets) is determined computably by the index i of ϕ i .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1807.02715v1">arXiv:1807.02715v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xa3u5f2xgjgh3jhw636olpm2mu">fatcat:xa3u5f2xgjgh3jhw636olpm2mu</a> </span>
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Solovay's theorem cannot be simplified

Andrew Arana
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤ T X of an "appropriate" Scott set and there  ...  is a family of functions (t n ) n∈ω , ∆ 0 n (X) uniformly in n, such that lim s→∞ t n (s) is an index for T ∩ Σ n and for all s, t n (s) is an index for a subset of T ∩ Σ n .  ...  Let A be a nonstandard model of PA, and let SS(A) = {d a : a ∈ A}. Then SS(A) is a Scott set. We call SS(A) the Scott set of A.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0168-0072(01)00095-1">doi:10.1016/s0168-0072(01)00095-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/utq4esovjbhqna5oeuzrevyhza">fatcat:utq4esovjbhqna5oeuzrevyhza</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20090902190510/http://www-personal.ksu.edu:80/~aarana/papers/final_solovay.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/df/96/df96723a383cb881332349a1dbcefcb36324ae15.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0168-0072(01)00095-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

The complexity of Scott sentences of scattered linear orders [article]

Rachael Alvir, Dino Rossegger
<span title="2019-09-16">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Given a countable scattered linear order L of Hausdorff rank α < ω_1 we show that it has a d-Σ_2α+1 Scott sentence. Ash calculated the back and forth relations for all countable well-orders.  ...  ., for every α < ω_1 there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable α the class of Hausdorff rank α linear orders is Σ_2α+2 complete.  ...  The complexity of the Scott sentence gives an upper bound on the complexity of the index set.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1810.11423v3">arXiv:1810.11423v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q4ohbfojzzdbxbj2acdfblqiwq">fatcat:q4ohbfojzzdbxbj2acdfblqiwq</a> </span>
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Complexity of Scott sentences

Rachael Alvir, Julia F. Knight, Charles McCoy CSC
<span title="">2020</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yeobluctzze7rfwekxs7sx2ayy" style="color: black;">Fundamenta Mathematicae</a> </i> &nbsp;
Miller (1983) , we show that if a countable structure A has a computable Σα Scott sentence and one that is computable Πα, then it has one that is computable d-Σ<α.  ...  We give effective versions of some results on Scott sentences.  ...  The first and second authors acknowledge support under NSF Grant #DMS 1800692.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/fm865-6-2020">doi:10.4064/fm865-6-2020</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eb254g5xgjb2zfcxg7jdcvze6i">fatcat:eb254g5xgjb2zfcxg7jdcvze6i</a> </span>
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Scott sentences for certain groups [article]

Julia Knight, Vikram Saraph
<span title="2017-07-07">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible.  ...  In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable d-Σ_2" (the conjunction of a computable Σ_2 sentence and a computable Π_2 sentence).  ...  This sentence is in fact a Scott sentence for F n . See [9] for the proof of this fact, along with the proof that the index set of F n is d-Σ 0 2 -hard.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1606.06353v2">arXiv:1606.06353v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ip2qniyr7rajrdiu66lc4pczru">fatcat:ip2qniyr7rajrdiu66lc4pczru</a> </span>
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Page 295 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 65, Issue 5 [page]

<span title="">1959</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_american-mathematical-society-bulletin" style="color: black;">American Mathematical Society. Bulletin of the American Mathematical Society </a> </i> &nbsp;
Frayne, Dana Scott, and Tarski, 1958 (see also S. Kochen, 1958), except in notation, let A? denote the ordinary direct product of algebras A‘, where 7 runs over an index set I.  ...  We conclude with a result of Log, 1955b, which has been reworked and improved by Frayne, Scott and Tarski, 1958 (also Morel, Scott, Tarski, 1958): P: all sentences are preserved under formation of reduced  ... 
<span class="external-identifiers"> </span>
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Classification from a Computable Viewpoint [article]

Wesley Calvert, Julia F. Knight
<span title="2008-03-22">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In model theory and descriptive set theory, there is a large body of work, showing that certain classes of mathematical structures admit classification, while others do not.  ...  Optimal Scott sentences and index sets Classification involves describing the structures in a class, up to isomorphism. One way to do this is by giving Scott sentences.  ...  For inspiration, we go to the setting of descriptive set theory, with arbitrary countable structures and Scott sentences in L ω1ω .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0803.3293v1">arXiv:0803.3293v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v64gbjnw4jghreao7w4nszzlxu">fatcat:v64gbjnw4jghreao7w4nszzlxu</a> </span>
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Not all narrative shifts function equally

Shannon Scott Rich, Holly A. Taylor
<span title="">2000</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ajltsigzdbc6zo6q2oqboml2gi" style="color: black;">Memory &amp; Cognition</a> </i> &nbsp;
In three experiments, we examined the role of basic event building blocks (character, time, and location) in event indexing during text comprehension.  ...  These three experiments dealt with perceived coherence, perceived cohesion, and on-line processing, respectively. The results indicated that characters are more likely to serve as event indexes.  ...  In both Taylor and Tversky (1997) and Taylor and Scott Rich (1995) , participants worked with a set of events defined by different instances of two event features.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3758/bf03211825">doi:10.3758/bf03211825</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/11126946">pmid:11126946</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tug4t3r3izeohoex4rbnywghmi">fatcat:tug4t3r3izeohoex4rbnywghmi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180727123612/https://link.springer.com/content/pdf/10.3758%2FBF03211825.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9d/e2/9de2d4609f6b2e78944b73f7f9d96084097326cf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3758/bf03211825"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Hanf Number for Scott Sentences of Computable Structures [article]

Sergey Goncharov, Julia Knight, Ioannis Souldatos
<span title="2016-11-01">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The same argument proves that _ω_1^CK is the Hanf number for Scott sentences of hyperarithmetical structures.  ...  Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is _ω_1^CK.  ...  Such a sentence is called a Scott sentence for A.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1602.01156v2">arXiv:1602.01156v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7bevsy3eijbvnizwuleduaqdx4">fatcat:7bevsy3eijbvnizwuleduaqdx4</a> </span>
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Classification from a Computable Viewpoint

Wesley Calvert, Julia F. Knight
<span title="">2006</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nxnim3i24fhcvkvgruk2dej7yu" style="color: black;">Bulletin of Symbolic Logic</a> </i> &nbsp;
We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures.  ...  Section 1 gives some background from model theory and descriptive set theory.  ...  Optimal Scott sentences and index sets Classification involves describing the structures in a class, up to isomorphism. One way to do this is by giving Scott sentences.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/bsl/1146620059">doi:10.2178/bsl/1146620059</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/efmtdpvde5e2tic5t4wunr4mbu">fatcat:efmtdpvde5e2tic5t4wunr4mbu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20060822172907/http://campus.murraystate.edu:80/academic/faculty/wesley.calvert/preprints/bsl.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a7/79/a7791c058feb815629ed589073699d5459234556.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2178/bsl/1146620059"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

An Introduction to the Scott Complexity of Countable Structures and a Survey of Recent Results

Matthew Harrison-Trainor
<span title="2021-11-15">2021</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nxnim3i24fhcvkvgruk2dej7yu" style="color: black;">Bulletin of Symbolic Logic</a> </i> &nbsp;
Such a sentence is called a Scott sentence, and can be thought of as a description of the structure.  ...  The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure.  ...  They also computed the complexity of the index sets of computable structures of Scott rank ω CK 1 and ω CK 1 + 1; these are complete for differences of Π 1 1 sets and differences of Σ 1 1 sets respectively  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/bsl.2021.62">doi:10.1017/bsl.2021.62</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/miy6gyratjdlflwar52e2c7tyq">fatcat:miy6gyratjdlflwar52e2c7tyq</a> </span>
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An introduction to the Scott complexity of countable structures and a survey of recent results [article]

Matthew Harrison-Trainor
<span title="2020-11-08">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Such a sentence is called a Scott sentence, and can be thought of as a description of the structure.  ...  The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure.  ...  The index set {i ∶ A i has high Scott rank} of computable structures with high Scott rank is Π 1 1 m-complete.  ... 
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